Physics for Solid State Applications

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6.730 Physics for Solid State Applications

Lecture 4: Vibrations in Solids

February 11, 2004

1-D Elastic Continuum

1-D Lattice Waves

3-D Elastic Continuum 3-D Lattice Waves

Outline

Strain E and Displacement u(x)

u(L)u(x+dx)u(x)

dx

dx

(dx) = dx dx how much does a differential length change

(dx) = u(x+dx) - u(x) difference in displacements

Strain:


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Uniform Strain & Linear Displacement u(x)

u(L)u(x+dx)u(x)

dx

dx

Constant Strain:

Linear displacement: u(x) = E0 x

More Types of Strain

u(L)u(x+dx)u(x)

dx

dxNon-uniform

E = E(x)

Zero Strain:

u(x) is constant

Just a Translation

We will ignore this


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11--D Elastic ContinuumD Elastic ContinuumStress and StrainStress and Strain

Lo

uniaxial loading

L

Stress:

Elongation: Normal Strain:

If ux is uniform there is no strain, just rigid body motion.

11--D Elastic ContinuumD Elastic ContinuumYoungs ModulusYoungs Modulus

CERAMICS GLASSES AND SEMICONDUCTORS

Diamond (C) 1000

Tungsten Carbide (WC) 450 -650

Silicon Carbide (SiC) 450Aluminum Oxide (Al2O3) 390

Berylium Oxide (BeO) 380

Magnesium Oxide (MgO) 250

Zirconium Oxide (ZrO) 160 - 241

Mullite (Al6Si2O13) 145

Silicon (Si) 107

Silica glass (SiO2) 94

Soda-lime glass (Na2O - SiO2) 69

METALS :

Tungsten (W) 406

Chromium (Cr) 289

Berylium (Be) 200 - 289

Nickel (Ni) 214

Iron (Fe) 196

Low Alloy Steels 200 - 207

Stainless Steels 190 - 200

Cast Irons 170 - 190

Copper (Cu) 124

Titanium (Ti) 116

Brasses and Bronzes 103 - 124

Aluminum (Al) 69

PINE WOOD (along grain): 10

POLYMERS :

Polyimides 3 - 5

Polyesters 1 - 5

Nylon 2 - 4

Polystryene 3 - 3.4

Polyethylene 0.2 -0.7

Rubbers / Biological

Tissues 0.01-0.1

Youngs Modulus For Various Materials (GPa)from Christina Ortiz


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Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave EquationD Wave Equation

Net force on incremental volume element:


Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia


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Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave Equation SolutionsD Wave Equation Solutions

Clamped Bar: Standing Waves

Periodic Boundary Conditions: Traveling Waves

Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave Equation SolutionsD Wave Equation Solutions


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33--D Elastic ContinuumD Elastic ContinuumVolume DilatationVolume Dilatation

Lo Lapply load

Volume change is sum of all three normal strains

33

--D Elastic ContinuumD Elastic Continuum

Poissons RatioPoissons Ratio

is Poissons Ratio ratio of lateral strain to axial strain

Poissons ratio can not exceed 0.5, typically 0.3


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Aluminum: EY=68.9 GPa, = 0.35

20mm

75mm5kN

5kN

33--D Elastic ContinuumD Elastic ContinuumPoissons Ratio ExamplePoissons Ratio Example


20mm

75mm5kN

5kN

33


Poissons Ratio ExamplePoissons Ratio Example


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20mm

75mm5kN

5kN

33--D Elastic ContinuumD Elastic ContinuumPoissons Ratio ExamplePoissons Ratio Example

33


Shear StrainShear Strain

2

Shear loading Shear plus rotation Pure shear

Pure shear strain

Shear stress

Gis shear modulus


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33--D Elastic ContinuumD Elastic ContinuumStress and Strain TensorsStress and Strain Tensors

For mostgeneral isotropic medium,

Initially we had three elastic constants: EY, G, e

Now reduced to only two: ,

33


Stress and Strain TensorsStress and Strain Tensors

If we look at just the diagonal elements

Inversion of stress/strain relation:


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33--D Elastic ContinuumD Elastic ContinuumExample ofExample of UniaxialUniaxial StressStress

Lo

L


Net force on incremental volume element:

Total force is the sum of the forces on allthe surfaces


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Net force in the x-direction:


Finally, 3-D wave equation.


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Dynamics of 3Dynamics of 3--D ContinuumD ContinuumFourier Transform of 3Fourier Transform of 3--D Wave EquationD Wave Equation

Anticipating plane wave solutions, we Fourier Transform the equation.

Three coupled equations for Ux, Uy, and Uz.

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumDynamical MatrixDynamical Matrix

Express the system of equations as a matrix.

Turns the problem into an eigenvalue problem forthe polarizations of the modes (eigenvectors) andwavevectors q (eigenvalues).


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Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSolutions to 3Solutions to 3--D Wave EquationD Wave Equation

Transverse polarization waves:

Longitudinal polarization waves:

1. Dynamical Equation can be solved by inspection

2. There are 2 transverse and 1 longitudinal polarizations for each q

3. The dispersion relations are linear

4. The longitudinal sound velocity is always greater thanthe transverse sound velocity

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSummarySummary

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Physics for Solid State Applications